# Calculating AUC (linear and log-linear)

When performing non-compartmental analysis, the area under the concentration-time curve (AUC) is calculated to determine the total drug exposure over a period of time. Together with Cmax, these two parameters are often used to define the systemic exposure of a drug for comparison purposes. For example, in bioequivalence trials, the entire statistical analysis is based on the comparison between formulations of AUC and Cmax. While the mathematics involved in the calculation of AUC are simple, there are nuances to the methods that are often misunderstood. Hopefully I can review some of the key details here. You can also view my video on YouTube.

Although AUC can be calculated directly from primary PK parameters (CL and V), I will discuss only the numerical estimation of AUC using noncompartmental analysis techniques in this blog post.

#### Linear Trapezoidal Method

The linear trapezoidal method uses linear interpolation between data points to calculate the AUC. This method is required by the OGD and FDA, and is the standard for bioequivalence trials. For a given time interval (t1 – t2), the AUC can be calculated as follows:

$AUC = \frac{1}{2}(C_1+C_2)(t_2-t_1)$

In essence the first two terms calculate the average concentration over the time interval. The last piece (t1 – t2) is the duration of time. So the linear method takes the average concentration (using linear methods) and applies it to the entire time interval. When you sum all of the intervals together, you will arrive at the total exposure from the first time point to the last. If you then divide the total AUC by the total time elapsed, you will arrive at the “average” concentration of drug in the body over the total time interval.

#### Logarithmic Trapezoidal Method

The logarithmic trapezoidal method uses logarithmic interpolation between data points to calculate the AUC. This method is more accurate when concentrations are decreasing because drug elimination is exponential (which makes it linear on a logarithmic scale). For a given time interval (t1 – t2), the AUC can be calculated as follows:

$AUC = \frac{C_1-C_2}{ln(C_1)-ln(C_2)}(t_2-t_1)$

This method assumes that C1 > C2. The fraction represents the logarithmic average of the two concentrations. Just as with the linear method, the average concentration is multiplied by the time interval.

#### Linear-Log Trapezoidal Method

This is a combination of the first two methods and is also called “linear-up log-down”. When concentrations are increasing (as in the absorption phase), the linear trapezoidal method is used. When concentrations are decreasing (as in the elimination phase), the logarithmic trapezoidal method is used. This method is thought to be the most “accurate” because the linear method is the best approximation of drug absorption while logarithmic decline is best modeled by the logarithmic trapezoidal method during drug elimination.

#### Why are there different methods?

The following figure demonstrates how the linear trapezoidal method overestimates the AUC during the elimination phase. The blue line represents mono-exponential decline of a drug. Samples were drawn at 16 and 20 hours. The red line represents the linear trapezoidal methods estimation of drug concentrations. As you can plainly see, the red line is higher than the blue line suggesting overestimation by the linear trapezoidal method.

Linear Trapezoidal Method

The logarithmic trapezoidal method accurately estimates mono-exponential decline of drug concentrations. However, during an absorption phase, the logarithmic trapezoidal method can underestimate the exposure.

I hope you have a better understanding of how to calculate AUC using the different methods that are available. And I hope you understand the basis of these methods and the pitfalls and limitations of each.

1. Hans says:

Any comment on the (typical?) practice of replacement of low concentration values below the limit of quantification (BLQ) by zero or BLQ/2 before such calculations?

• Great question Hans. Instead of answering in the comments, I will prepare a separate blog post on the topic. Thank you for such an engaging question.

2. balaj says:

Calculation of volume of distribution.
Vd is calulated by formula Vd = Dose/Cp0 & by another formula Vd = CL/Kel, but answer by these two method does not some similar, why & help to solve the problem to get the same answer by these formulas.

• Thank you for the question. Assuming you have an IV dose, and a quality estimate of Cp0, I believe the best method for calculating Vd is the first equation you presented (Vd = Dose/Cp0). The second equation should really be turned around differently and used as follows: Kel = CL/Vd. The parameter Kel is a secondary pharmacokinetic parameter and derived from the primary parameters CL and Vd. If you try to use it in the method you describe, any inaccuracies in Kel will be magnified in the estimate of Vd leading to the discrepancy that you noted. I hope that helps.

• balaji says:

Thank you Nathan for explanation, still one doubt remains is that, in WinNonlin NCA, Vd is calculated by Vd = CL/Kel & not by the other equation i.e. Vd=D/Cp0. When we put the value of Cp0 calculated after NCA analysis in this formula answer for Vd not matches with WinNonlin output value of Vd. Please clarify.

• WinNonlin is designed to calculate Vd using the clearance and terminal elimination rate constant. The reason these values do not match is due to the variability in the estimation of both Cp0 and kel. Since these two values are calculated independently from different parts of the plasma concentration-time curve, any variance from the “true” values will result in differences in Vd estimates. If you are using mean data, then these differences in Vd estimates may be magnified. I would recommend a nonlinear model-based fit to accurately determine the volume of distribution.

• Dr. Imran says:

These two equations ll yield same results only when drug follows one compartment open model kinetics.

3. Hanna says:

There are some factors contributing to blood glucose level (BGL). How can I assess the influence of different doses and different insulin types (rapid and slow acting) while calculating AUC of BGL in type I diabetes patients that depend upon external insulin?

• Nathan Teuscher says:

4. JS says:

Can there be any situation where AUC(0-24) is greater than Cmax? I recently came across such data and want to be sure if that is a mistake.

• Nathan Teuscher says:

AUC(0-24) and Cmax are different parameters with different units. AUC(0-24) is a measure of exposure over time and has units of concentration*time (e.g. ng*h/mL). Cmax is a measure of peak exposure at a specific time point and has units of concentration (e.g. ng/mL). Therefore you cannot directly compare numbers across parameters. Perhaps you can share more information about the specific scenario so I can help answer the question.

-Nathan

5. UT says:

I have a IV 2-compt. PK profile. I am interested in calculating the distribution half-life ( alpha half-life) using non-compartmental analysis. For performing the computation, I am using the Phoenix Winnonlin software package. In the setup of non-compartmental module, I am selecting the distribution phase instead of the terminal elimination phase in the slope selector section.

I was wondering that by doing so am I violating any assumptions for non-compartmental analysis as we usually need to select the terminal phase for lamda z calculation?

• Nathan Teuscher says:

Simply selecting the timepoints during the “distribution phase” will not give an accurate representation of the distribution half-life. Simply selecting the early time points will provide an estimate of a rate constant that is a combination of the distribution and elimination processes. I would recommend performing curve stripping instead using Excel or some other spreadsheet program. You can refer to a previous post I made on this topic (link). You can also perform curve stripping with Phoenix WinNonlin by performing a model fit to the profile.

6. Tan says:

For toxicokinetic studies, we don’t always get all the time points needed to get a nice curve. In some cases, we may get 1 or 2 time points with measurable values, but the remaining part of the curve is BLQ. What is the minimum number of measurable concentration data needed to calculate AUC? What is the standard practice?

• Nathan Teuscher says:

I don’t know of a standard practice; however, my opinion is that at least 5 data points are necessary to calculate an AUC value. Although as the number of data points drops, the confidence should also drop that the AUC actually represents total exposure.

7. Kristina says:

Hi Nathan
I am interested in calculating AUC from 4 measurements; the first one before drug administration (on the second day after initiation of treatment), the following ones 1, 3 and 24 hours later. The medicine is administered once daily so the last measurement is taken right before the next dose is given. Cmax is reached approximately after one hour so I am thinking linear trapezoidal method for the first 2 measurements and logarithmic trapezoidal method for the rest. Does this make sence? I know 4 points are not a lot, but I am very interested in getting an AUC value.

• Nathan Teuscher says:

Kristina,

You can calculate an AUC from 4 concentration values; however, it may be highly variable. I would recommend using linear trapezoidal when concentrations are increasing and logarithmic when concentrations are decreasing. Assuming that drug increases from 0-1 hr and then decreases from 1-24 hr, then your suggested plan is appropriate.

Nathan

• Kristina says:

Thank you for your answer! I am measuring concentrations of Moxifloxacin, and my problem is that for some patients, the concentration is still increasing at 3 hours. For these patients I have increasing concentration from 0-3 hours and then of course decreasing concentration at 24 hours. I am thinking that Peak/MIC might be a better estimate to use for analysing data than AUC/MIA with such few measurements. What is your opinion on that?

Kristina

• Nathan Teuscher says:

The method you use to calculate AUC does not really affect the PK/PD relationships you are trying to make. I would actually recommend developing a population PK model to correlate with efficacy rather than noncompartmental analysis. There are many published articles on the population pharmacokinetics of moxifloxacin. Then you can use those concentrations to investigate pharmacodynamic responses.

• Kristina says:

With the risk of sounding stupid, can WinNonlin be used for developing a population PK model? I ask because I am currently looking into which program to use for PK/PD calculations.
I am interested in finding out whether peak concentration of Moxifloxacin 400 mg daily reaches >10 times the MIC for common bacteria causing community-acquired pneumonia (using MIC from EUCAST).

• Nathan Teuscher says:

Not a stupid question at all. Phoenix WinNonlin cannot be used to develop a population PK model. However, Phoenix NLME can be used. I write about the NLME package in another post that you can read here. The NLME package requires an additional license from Pharsight/Certara.

• Kristina says:

I find it hard to find guidelines on how to create a population PK model, like how many subjects to use. Is there a golden standard?

I´ve read what you wrote about Phoenix NLME. Would you recomend this program rather than NONMEM? And what about online training for a beginner like myself? Is there an online course for Phoenix NLME like there is for Phoenix WinNonlin?

• Nathan Teuscher says:

Kristina,
There is no “golden standard” for the number of subjects required to conduct a population PK analysis. The more data that you have, the more precise and accurate your model becomes. I use both Phoenix NLME and NONMEM. In my opinion, Phoenix NLME is easier for most people who are new to population PK anlaysis, and it appears to be just as powerful as NONMEM. NONMEM has a longer history, and can be more flexible in certain situations, but it requires more effort to learn how to use the software and related tools to help with data formatting and figure creation. I have prepared an online course Population PK for Beginners that provides an introduction to the methods and software used for analysis. I show how to use NONMEM and Phoenix NLME to perform basic population PK analysis.
Nathan

8. Thomas says:

I know using AUC 0-infi. is optimal in single dosed studies and that using AUC 0-Tau is optimal in multidosed studies. Is this because using AUC 0-infinity in multidosed studies overestimates the AUC?

Similarly, when calculating AUC from Clearance using the equation 1/AUC=CL does this equate to AUC 0 to infinity?

• Nathan Teuscher says:

Thomas,

AUC0-inf following a single dose represents the total drug exposure after one dose of study drug. Clearance can be calculated using the dose and AUC0-inf using:
CL = Dose/AUC0-inf

Some people use AUC0-t in place of AUC0-inf; however, AUC0-t is always less than AUC0-inf (because they haven’t extrapolated to infinity) therefore, CL estimates with AUC0-t are slightly high.

When administering multiple doses, there are unique properties when you reach steady-state. At steady state, AUC0-tau (at steady-state) = AUC0-inf (single dose). This occurs because drug from previous doses is still present and measured as part of AUC0-tau. Thus if you are at steady-state, you can calculate Clearance using the following equation:
CL = Dose/AUC0-tau

AUC0-inf from the last dose of a multiple dose study is not useful. Why? Because it includes the total exposure from the last dose AND previous doses. Therefore it does not provide meaningful information. I do not recommend calculation of AUC0-inf in multiple-dose studies for this reason.

Nathan

• Thomas says:

Thank you! I’m definitely going to take your course.

9. Karin says:

Hi,

I wonder if FDA still recommends using the linear trapezoidal method for calculating AUC.

• Nathan Teuscher says:

The OGD (Office of Generic Drugs) at the FDA still recommends the linear trapezoidal method in their guidance documents. The other divisions of the FDA do not specify an AUC method in their guidance documents.

• Alice says:

Hi Nathan,
Can you please offer the name (or link) of the guidance doc the OGD states this recommendation? I cannot seem to find it.
Thank you!

• Nathan Teuscher says:

The recommendation is based on personal experience with the OGD several years ago. There is no formal guidance on the methods used to calculate the AUC, and as such, the OGD may have changed their opinion.

Nathan

10. Annie says:

Hi Nathan,

Thank you for the excellent presentation and I found it very informative and useful.

This may sound silly but I cannot seem to reproduce the math on lnC1-lnC2 and wonder how exactly is this being calculated?

• Nathan Teuscher says:

Annie,

The derivation is very long (took me over 2 pages), so I didn’t include it on this post. You can verify that it works by calculating the log trapezoidal rule for a set of points and then transforming the concentrations to their logarithmic equivalents and calculating the linear trapezoidal rule (don’t forget to back-transform into standard units). You should get the same answer. Essentially, you are calculating the area of a trapezoid in two different coordinate systems.

Nathan

11. galen says:

hey, how do i calculate shelf life of pasteurized juice having microbial results

• Nathan Teuscher says:

Galen,

I am not an expert in calculating shelf-life for food products. Sorry.

Nathan

12. Zora says:

Hi,
A drug is given as per the labeling in 3 divided doses and where the max time between doses should not exceed 12 hours”. So it can be given 8-8-8 or 6-6-12, or otherwise, as long a the longest interval is <=12.
If we give 3 doses Q6 hrs during the day and then none at night, so the intervals BTW the 3 doses are 6, 6, 12. Which tau shall we use for AUCtau?
Thank you.
Best regards,
Zora

• Nathan Teuscher says:

Zora,

If you have unequal intervals, then you will have different time windows for each AUCtau. The “tau” represents the time between successive doses, so between Dose 1 and 2, tau would be 6 hours; between Dose 2 and 3, tau would be 6 hours; and between Dose 3 and Dose 1 the next day, tau would be 12 hours.

Nathan

• Zora says:

Thank you very much Nathan.

Sorry if you get this question more than once. I am having problems with the site and am not sure if my reply was sent. THe page froze, so I am not sure.

So on the last dosing day the doses are at 0, 6, an 12 hrs, so the intervals are 6, 6, and 12. The last dosing interval has a 12-hour sampling (Hours 12-24 on the last day), so the AUC tau is AUC12-24 (tau-12).

But were you saying that there I should also consider the tau= 6 for the first two doses on that day: AUC0-6 and AUC6-12 ? last day ? And have 3 AUCtaus reported?

Thank you very much again, appreciate your help.

Best regards,

Zora

• Nathan Teuscher says:

Yes, your interpretation is correct. You should capture the 3 AUCtaus listed in your comment.

13. vardhaman says:

Dear Nathan,

In a study , i got results wherein for a volunteer , Cmax was found to be around 10800 ng/ml where as AUC 0-24 hrs for the same volunteer for the same treatment was found to be around 10600 ng/ml.
Generally, I presume, AUC should always be higher than the Cmax. However, this is a rare observation, is this due to some bioanalytical or calculation error.

Can you pl. share your views.

• Nathan Teuscher says:

Vardhaman,

I’m not sure I understand the question. The units for AUC should be in time-concentration. Examples include: ng*hr/mL or ug*min/mL. In your question, the AUC0-24 value has units of ng/mL, which is not correct. Perhaps you could provide more information about the methods used to calculate the values.

Nathan

14. Kam says:

Hi,
I have question regarding calculating total exposure(or total amount of drug) to subject when we have only data regarding daily dose taken like Day 1: 400mg, Day2:400mg,Day3:800mg,Day4:0mg,Day5:400mg likewise. One way is to calculate by direct summation and using it. However, can we use by ploting it on graph and from AUC using Trapezoid rule? Would it make difference or it would be same as summation? I know (direct summation), but again I am asking, which one is more meaningful from your perspective (since we have discussion over this with knowledgeable person)? What if difference comes due to mathematical calculation from trapezoid vs. direct summation. Thanks a lot in advance.
Kam

• Nathan Teuscher says:

Hi Kam,

I’m not sure I understand your question. Total exposure (or area under the concentration-time curve or AUC) is calculated by using the concentration-time data. You cannot calculate AUC using only the dose unless you have a validated PK model derived from previous concentration-time data. The trapezoidal method for AUC calculations is a summation method (you sum up all the small trapezoids). I’m unclear what “direct summation” is.

Nathan

• Kam says:

Hi Nathan,
Let me make my question more clear. First of all, we are not looking into blood-concentration data. When I say, exposure, I refer it to amount of drug taken by subject. It is very straight forward to get total amount of drug taken (i.e. here exposure) by summing all the dose subject has taken. This is how we are doing it. Now, we have been proposed for another method of plotting Daily Dose taken in mg (Y axis) Vs. Time in day plot and calculating AUC for that plot to calculate total amount of dose taken by subject. I agree, it is indirect method of calculating total amount of dose taken when we have actual dose taken data in our hand. However, our hypothesis is if subject has discontinued drug or interrupted for some days, it would have different answers. For example, when total dose is zero on day 15 (when there is interruption), we take it zero. But, in graphical method (AUC using trapezoid rule), the curve would come to zero, which would create a triangle. This is the difference. So, can you please give me some insights/thoughts on this. When it would be more appropriate to use AUC method vs. simple summation of dose taken.
Just to note again, we are concerned with amount of dose taken by subject and it is what we say “exposure” here (Not the blood concentration). I understand, Pharmacokinetically blood exposure is calculated using AUC method and it implies the amount of dose taken.
Kam

• Nathan Teuscher says:

Kam,

Thank you for the clarification. I think I understand better now. I have never performed an analysis such as the one you are describing. I think that the direct summation method (adding up total dose administered) will be more accurate than using a trapezoidal rule to calculate an area. Using the example you stated, the subject took 2000 mg (400 + 400 + 800 + 0 + 400) over 5 days. But if you used the trapezoidal method, and set the dose at Day 0 to 0 mg, you would get 1600 mg. I’m not sure that the logic behind using an AUC-type analysis makes sense with dose data. AUC is used when the y variable is continuously changing and is a surrogate for integration. In this case, the amount of drug administered is known perfectly, so you don’t need to integrate a curve. I would recommend using direct summation in this case.

I hope that helps!
Nathan

• Kam says:

Hi Nathan,
Thanks a lot.
It is more helpful and clear. This what actually I wanted to know when we use AUC (as you said when continuously changing).
Have a nice day!

Kam

15. HG says:

Hi Nathan,

On the topic of ‘Calculation Method’ in WinNonlin, there are two options called ‘Linear Trapezoidal Linear Interpolation’ and ‘Linear Trapezoidal Linear/Log Interpolation’ that both appear to calculate AUC using the linear trapezoidal method, but will presumably calculate Lamda z differently. Would you please explain the difference and the circumstances under which each method is appropriate?

• Nathan Teuscher says:

Hi,

The difference between the “Linear Trapezoidal Interpolation” and the “Linear Trapezoidal Linear/Log Interpolation” methods are related to implementation. The linear trapezoidal interpolation method uses the linear trapezoidal method to calculate AUC for all time points. If a partial area is required for which a time point doesn’t exist (e.g. AUC 6-8 hrs and you have samples at 6 and 10 hrs), the method will use linear interpolation between the two observed data points to estimate a predicted concentration at 8 hrs that will be used in the partial AUC calculation. The linear trapezoidal linear/log interpolation method also uses linear trapezoidal method to calculate the AUC for all time points. If a partial area is required for which a time point doesn’t exist, the method will use linear interpolation for increasing concentrations and logarithmic interpolation for decreasing concentrations.

None of these methods affect the estimation of Lambda Z.

I generally prefer the linear up/log down method over the two you indicated. There is a small difference between linear and logarithmic AUC calculations for decreasing concentrations, which is unlikely to cause any significant difference in total AUC in most cases. The linear AUC calculation slightly over-estimates the AUC compared to the logarithmic AUC method, but the difference is rarely observable unless the time windows and/or the concentration differences are large between consecutive data points.

I hope that helps!
Nathan

• HG says:

Nathan – Thank you so much for your very detailed response. This was very helpful.

I am attempting to calculate the most conservative estimate of AUC for this project, but will definitely keep in mind your recommendation of the linear up/log down approach for future projects.

16. Bjoern Bauer says:

Hi Nathan,
From briefly scanning over the previous messages, I was wondering if there is somewhere a comparison/correlation of the trapezoidal method (estimating AUC) vs. methods that actually calculate AUC (integration)? How close does the estimation come to the “real” AUC? How far off is it (or not)? My apologies if I am repeating something that has been posted before.
Thanks,
Bjoern

• Nathan Teuscher says:

Bjoern,

Differences between numerically integrated AUC and model-estimated AUC depend on the variability of the data and quality of the model fit. Since AUC is a “calculated value”, there isn’t a “real” AUC … just estimation methods. Numerical integrations are used when a model is not available, model-estimated AUCs are used when a PK model exists to describe the data.

Nathan

17. Rachel says:

Hi Nathan,

I would like to know which is appropriate to use for a hepatic portal vein dose–extravascular model or IV bolus model? I know the dose is administered into a vein, but the first pass effect is still present, so I am not sure which model is correct?

Thanks,
Rachel

• Nathan Teuscher says:

Rachel,

When you administer to a portal vein, you don’t have an absorption rate constant, but you have a bioavailability factor. I think you will need a custom model to fit that data. You could use an IV model, and add a bioavailability parameter. Unless you have a true IV dose there is no way to calculate the absolute bioavailability. Thus, without a true IV dose you can simply use an IV model and report CL/F and V/F as your parameters.

Nathan

• Rachel says:

Nathan, thank you for your response. For this study, I am most interested in measuring the Cmax and AUC with and without coadministration of an inhibitor.

Thanks,
Rachel

• Nathan Teuscher says:

Cmax is independent of route of administration. The only difference between route of administration for calculation of AUC is how the first data point is handled. In extravascular, concentration is assumed to be zero at time = 0. For IV, the concentration at time = 0 is estimated by back-extrapolation to the origin. I guess it would depend on your data … if concentrations start high and fall over time then I would use IV. If concentrations rise from a low value to a peak and then fall, I would use extravascular.

• Rachel says:

Great, this was my understanding, so thank you for verifying. In this case, the plasma concentrations decline like an IV bolus so I will use this model. I would also like to measure the metabolite, in which case I suppose the extravascular model would be best since the levels rise and then fall?

• Nathan Teuscher says:

Yes. That would be the best model for the metabolite.

• Rachel says:

18. Alexandra says:

Hi Nathan,

to calculate the dose of a benzodiazepine a patient is intoxicated with I have two plasma concentrations and the approximated hours regarding timepoint 0.
I am not quite sure which approach is the correct one:
Either I calculate AUC0-∞ and then use the formula Cl = (Foral*dose)/AUC0-∞
Or I use this dose: C(t) = (Dose/V) * (e-kel*t)

My apologies if any of my assumptions is completely wrong.
Thank you and best regards, Alexandra

• Nathan Teuscher says:

Both equations can work if you have all of the necessary parameters. The first equation requires that you have the oral bioavailability (Foral), total exposure (AUC0-inf) and the clearance (Cl). The second equation requires that you have the volume of distribution (V) and elimination rate constant (kel). The second equation assumes IV administration, if it is oral administration, additional variables are needed.

The best option would be to use a known PK model for benzodiazepine and input the concentrations and solve for the dose.

Nathan

• Alexandra says:

Thank you very much, Nathan.

19. slk says:

Hi Nathan,
I am wondering if AUC is always underestimated during ascending concentration? Could the ascending concentration curve be concave and hence AUC will be overestimated?

• Nathan Teuscher says:

It is a possibility.

• slk says:

Thank you for your reply. Could you also explain why “during an absorption phase, the logarithmic trapezoidal method can underestimate the exposure”? I could not visualize how the curves look like. Also, which method do you think is the best to estimate AUC, “linear up log down” or “linear log trapezoidal”? The standard practice that I have known all along is always “linear log trapezoid”. But recently I encounter PK profiles that are going up and down and “linear up log down” method gives higher AUC estimate that “linear log trapezoid”. Thank you for your help!

• Nathan Teuscher says:

Generally logarithmic method has a curvature to it that is concave on the linear scale. The absorption phase generally is linear or convex on the linear scale. Thus the logarithmic method will underestimate the concentrations. I don’t have a preference for which method is the “best”. It is important to understand the features of each method. Both the linear method and linear up/log down method are commonly used.